Question

If you mix \(48 \mathrm{mL}\) of \(0.0012 \mathrm{M} \mathrm{BaCl}_{2}\) with \(24 \mathrm{mL}\) of \(1.0 \times$$10^{-6} \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) will a precipitate of \(\mathrm{BaSO}_{4}\) form?

Short answer

Yes, a precipitate will form because the ionic product is greater than the solubility product.

Step-by-step solution

Write the Balanced Equation

The reaction between barium chloride and sodium sulfate can be written as \(\mathrm{BaCl}_2 (aq) + \mathrm{Na}_2 \mathrm{SO}_4 (aq) \rightarrow \mathrm{BaSO}_4 (s) + 2\mathrm{NaCl} (aq)\). Here, barium sulfate \(\mathrm{BaSO}_4\) is an insoluble salt, so if the concentration product reaches its solubility product, a precipitate will form.

Calculate Initial Concentrations After Mixing

After mixing, the total volume is \(48 \mathrm{mL} + 24 \mathrm{mL} = 72 \mathrm{mL}\).The initial concentration of \(\mathrm{Ba}^{2+}\) will be:\[C_{\mathrm{Ba}^{2+}} = \left(48 \mathrm{mL} \times 0.0012 \mathrm{M}\right) / 72 \mathrm{mL} = 0.0008 \mathrm{M}\]The initial concentration of \(\mathrm{SO}_4^{2-}\) will be:\[C_{\mathrm{SO}_4^{2-}} = \left(24 \mathrm{mL} \times 1.0 \times 10^{-6} \mathrm{M}\right) / 72 \mathrm{mL} = 3.33 \times 10^{-7} \mathrm{M}\]

Calculate Ionic Product (IP)

To determine if a precipitate will form, we calculate the ionic product \(\text{IP}\):\[\text{IP} = [\mathrm{Ba}^{2+}][\mathrm{SO}_4^{2-}] = (0.0008)(3.33 \times 10^{-7}) = 2.664 \times 10^{-10}\]

Compare Ionic Product to Solubility Product (Ksp)

The solubility product constant \(\text{K}_{sp}\) for \(\mathrm{BaSO}_4\) is approximately \(1.1 \times 10^{-10}\). Since \(\text{IP} = 2.664 \times 10^{-10} > \text{K}_{sp} = 1.1 \times 10^{-10}\), a precipitate of \(\mathrm{BaSO}_4\) will form.

Key concepts

Solubility Product

When a compound is only slightly soluble in a solvent, it reaches an equilibrium between the dissolved ions and the undissolved solid. This is known as the solubility product (Ksp). For any given temperature, the Ksp of a compound is a constant that signifies the level at which that compound's ions will dissolve in a solution to form an equilibrium state.

For barium sulfate (BaSO_4), its Ksp represents the maximum product of the concentrations of its ions (Ba^{2+} and SO_4^{2-}) that can occur before precipitate must form to maintain equilibrium.

The equation for this is given by:
\[K_{sp} = [Ba^{2+}][SO_4^{2-}]\] where \([Ba^{2+}]\) and \([SO_4^{2-}]\) are the concentrations of the barium and sulfate ions in a saturated solution.

• Higher Ksp means a compound is more soluble.
• Lower Ksp leads to less solubility, more likely to form a precipitate.

Ionic Product

The ionic product (IP) is similar to the solubility product, but it is calculated for any given solution, regardless of saturation. It helps determine whether a precipitate will form in a chemical reaction.

The IP is calculated using the concentrations of the ions at any point in the reaction. If the solution is not saturated, the IP can vary, while the Ksp remains constant for a specific substance at a specific temperature:
\[\text{IP} = [Ba^{2+}][SO_4^{2-}]\] The relationship is crucial for identifying precipitation:

• If IP < K_{sp}: The solution can dissolve more ions; no precipitate forms.
• If IP = K_{sp}: The solution is at equilibrium and perfectly saturated; no change occurs.
• If IP > K_{sp}: The solution is supersaturated; a precipitate begins to form to achieve equilibrium once more.

Barium Sulfate

Barium sulfate ( BaSO_4 ) is an inorganic compound that is characterized by its very low solubility in water. This property makes barium sulfate particularly useful in applications where it is beneficial for the compound to remain largely undissolved.

In precipitation reactions, like in the mixing of barium chloride and sodium sulfate, barium sulfate takes notable importance because of its distinct ability to precipitate out of the solution when its ions exceed its solubility limits.

• Barium sulfate is commonly used in medical imaging due to its opacity to X-rays.
• It is also valuable in industrial applications like oil well drilling.

In chemical equilibrium, the presence of barium sulfate as a solid in a reaction indicates that the equilibrium has likely been reached or surpassed, causing excess ions to be removed from the solution in the form of a precipitate.

Chemical Equilibrium

Chemical equilibrium in the context of a precipitation reaction signifies a state where the rate at which a solid dissolves into ions equals the rate at which the ions combine to form a solid precipitate. This dynamic balance makes exactly enough ions dissolve or come out of the solution to keep the solution saturated.

For reactions involving insoluble salts like barium sulfate, equilibrium is crucial. It determines whether more salt can dissolve or if solid will precipitate:

• If a solution begins with low ion concentrations and they increase, reaching the Ksp, saturation is achieved, bringing the system to equilibrium.
• If ion concentrations exceed the Ksp, equilibrium prompts a reaction that leads to the precipitation.

Chemical equilibrium thus ensures that excess ions are removed when needed to sustain the most balanced state, optimizing for solubility constraints.